Factorial designs factor

ED (BGE optimization). Three-level fnll factorial design. Factors SDS concentration, pH, phosphate concentration. Response resolntion and migration time. 

ED (BGE optimization). Two-level full factorial design. Factors buffer concentration, organic solvent, injection time, voltage, temperature. Response resolution, efficiency, tailing factor, migration time. 

Figure 22.7 The response surface for the steam temperature predicted using a T factorial design. Factor 1 (xi) ratio of water vaporized in the electrical vaporizer to the total amount of water. Factor 2 (X2) ratio of cold air fed into the reformer to the total amount of air. Factor 3 (xj) fuel flow rate kept constant at its upper value (1).

Factorial design methods cannot always be applied to QSAR-type studies. For example, i may not be practically possible to make any compounds at all with certain combination of factor values (in contrast to the situation where the factojs are physical properties sucl as temperature or pH, which can be easily varied). Under these circumstances, one woul( like to know which compounds from those that are available should be chosen to give well-balanced set with a wide spread of values in the variable space. D-optimal design i one technique that can be used for such a selection. This technique chooses subsets o... 

The linear regression calculations for a 2 factorial design are straightforward and can be done without the aid of a sophisticated statistical software package. To simplify the computations, factor levels are coded as +1 for the high level, and -1 for the low level. The relationship between a factor s coded level, Xf, and its actual value, Xf, is given as... 

A 2 factorial design with two factors requires four runs, or sets of experimental conditions, for which the uncoded levels, coded levels, and responses are shown in Table 14.4. The terms Po> Po> Pfc> and Pafc in equation 14.4 account for, respectively, the mean effect (which is the average response), first-order effects due to factors A and B, and the interaction between the two factors. Estimates for these parameters are given by the following equations... 

Example of Uncoded and Coded Factor Levels and Responses for a 2 Factorial Design... 

The computation just outlined is easily extended to any number of factors. For a system with three factors, for example, a 2 factorial design can be used to determine the parameters for the empirical model described by the following equation... 

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

Curved one-factor response surface showing (a) the limitation of a 2 factorial design for modeling second-order effects and (b) the application of a 3 factorial design for modeling second-order effects. 

If the actual response is that represented by the dashed curve, then the empirical model is in error. To fit an empirical model that includes curvature, a minimum of three levels must be included for each factor. The 3 factorial design shown in Figure 14.13b, for example, can be fit to an empirical model that includes second-order effects for the factor. 

In general, an -level factorial design can include single-factor and interaction terms up to the ( - l)th order. 

Duarte and colleagues used a factorial design to optimize a flow injection analysis method for determining penicillin potentiometricallyd Three factors were studied—reactor length, carrier flow rate, and sample volume, with the high and low values summarized in the following table. 

Here is a challenge McMinn and co-workers investigated the effect of five factors for optimizing an H2-atmosphere flame ionization detector using a 2 factorial design. The factors and their levels were... 

Factor Analysis, 78 Factor spaces, 78 Factorial design, 29 Factors, 94... 

As an analytical method becomes more complex, the number of factors is likely to increase and the likelihood is that the simple approach to experimental design described above will not be successful. In particular, the possibility of interaction between factors that will have an effect on the experimental outcome must be considered and factorial design [2] allows such interactions to be probed. 

Factorial design One method of experimental design that allows interactions between factors to be investigated, i.e. whether changing one experimental variable changes the optimum value of another. 

Based on the experimental data kinetic parameters (reaction orders, activation energies, and preexponential factors) as well as heats of reaction can be estimated. As the kinetic models might not be strictly related to the true reaction mechanism, an optimum found will probably not be the same as the real optimum. Therefore, an iterative procedure, i.e. optimization-model updating-optimization, is used, which lets us approach the real process optimum reasonably well. To provide the initial set of data, two-level factorial design can be used. 

As an example, a seven factor quarter fraction factorial design would be written as seen in equation below... 

Expediently, factorial design is done on the basis of transformed factors Zi, calculated from the x by... 

Because the experimental expenditure increases strongly with the increasing number of influence factors, fractional factorial design FFD (partial factorial design) is applied in such cases. It is not possible to evaluate all the interactions by FFDs but only the main effects. 

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